Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $401$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,16,8,5,14,11)(2,15,7,6,13,12)(3,17,10,4,18,9), (1,18,9,3,15,11,6,14,7)(2,17,10,4,16,12,5,13,8), (1,12,15,4,7,13)(2,11,16,3,8,14)(5,9,17,6,10,18) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 12: $A_4$ x 5 24: $A_4\times C_2$ x 5 48: $C_2^4:C_3$ 96: 12T56 648: $S_3 \wr C_3 $ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4\times C_2$
Degree 9: $S_3 \wr C_3 $
Low degree siblings
18T399 x 12, 18T401 x 11Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy Classes
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,17,16)(14,18,15)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $12$ | $3$ | $( 7,11, 9)( 8,12,10)(13,16,17)(14,15,18)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 1, 3, 6)( 2, 4, 5)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 6, 6, 3, 3 $ | $24$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7,10,11, 8, 9,12)(13,18,16,14,17,15)$ |
| $ 6, 6, 1, 1, 1, 1, 1, 1 $ | $12$ | $6$ | $( 7,12, 9, 8,11,10)(13,15,17,14,16,18)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $6$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $6$ | $( 7, 8)( 9,10)(11,12)(13,18,16,14,17,15)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $12$ | $6$ | $( 1, 6, 3)( 2, 5, 4)( 7,12, 9, 8,11,10)(13,14)(15,16)(17,18)$ |
| $ 6, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $6$ | $( 7,12, 9, 8,11,10)(13,14)(15,16)(17,18)$ |
| $ 6, 3, 3, 2, 2, 2 $ | $12$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7, 8)( 9,10)(11,12)(13,18,16,14,17,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 3, 6)( 4, 5)(15,18)(16,17)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $54$ | $6$ | $( 1, 3)( 2, 4)( 7, 9,11)( 8,10,12)(13,17)(14,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $27$ | $2$ | $( 3, 6)( 4, 5)( 7, 8)( 9,10)(11,12)(13,14)(15,17)(16,18)$ |
| $ 6, 2, 2, 2, 2, 2, 1, 1 $ | $54$ | $6$ | $( 1, 3)( 2, 4)( 7,10,11, 8, 9,12)(13,18)(14,17)(15,16)$ |
| $ 6, 2, 2, 2, 2, 2, 1, 1 $ | $54$ | $6$ | $( 1, 4)( 2, 3)( 5, 6)( 7,10,11, 8, 9,12)(13,17)(14,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,10)(11,12)(15,18)(16,17)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 2 $ | $54$ | $6$ | $( 1, 4)( 2, 3)( 5, 6)( 7, 9,11)( 8,10,12)(13,18)(14,17)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 1, 2)( 3, 5)( 4, 6)(13,14)(15,17)(16,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $144$ | $3$ | $( 1, 8,14)( 2, 7,13)( 3,10,18)( 4, 9,17)( 5,11,16)( 6,12,15)$ |
| $ 9, 9 $ | $288$ | $9$ | $( 1,12,18, 3, 8,15, 6,10,14)( 2,11,17, 4, 7,16, 5, 9,13)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $144$ | $3$ | $( 1,14, 8)( 2,13, 7)( 3,18,10)( 4,17, 9)( 5,16,11)( 6,15,12)$ |
| $ 9, 9 $ | $288$ | $9$ | $( 1,15,10, 3,14,12, 6,18, 8)( 2,16, 9, 4,13,11, 5,17, 7)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $81$ | $2$ | $( 3, 6)( 4, 5)( 9,11)(10,12)(13,14)(15,17)(16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $27$ | $2$ | $( 1, 4)( 2, 3)( 5, 6)( 7,10)( 8, 9)(11,12)(13,18)(14,17)(15,16)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 9,11)(10,12)(13,14)(15,16)(17,18)$ |
| $ 6, 3, 3, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9)( 8,10)(13,18,16,14,17,15)$ |
| $ 6, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 7,11)( 8,12)(13,15,17,14,16,18)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 1, 1 $ | $18$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 9,11)(10,12)(13,14)(15,16)(17,18)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 7, 8)( 9,12)(10,11)$ |
| $ 3, 3, 3, 3, 2, 2, 2 $ | $36$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7,10)( 8, 9)(11,12)(13,17,16)(14,18,15)$ |
| $ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 7,12)( 8,11)( 9,10)(13,16,17)(14,15,18)$ |
| $ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7, 8)( 9,12)(10,11)$ |
| $ 6, 6, 2, 2, 2 $ | $36$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,10)( 8, 9)(11,12)(13,18,16,14,17,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,18)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $18$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,18)$ |
| $ 6, 2, 2, 2, 2, 2, 2 $ | $18$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,12)( 8,11)( 9,10)(13,15,17,14,16,18)$ |
| $ 6, 3, 3, 2, 2, 1, 1 $ | $36$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7, 9)( 8,10)(13,17,16)(14,18,15)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 9,11)(10,12)$ |
| $ 6, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 9,11)(10,12)$ |
| $ 3, 3, 2, 2, 2, 2, 2, 1, 1 $ | $18$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,11)( 8,12)(13,16,17)(14,15,18)$ |
| $ 6, 6, 6 $ | $432$ | $6$ | $( 1, 8,14, 2, 7,13)( 3,12,18, 5, 9,16)( 4,11,17, 6,10,15)$ |
| $ 6, 6, 6 $ | $432$ | $6$ | $( 1,14, 7, 2,13, 8)( 3,15, 9, 5,17,12)( 4,16,10, 6,18,11)$ |
Group invariants
| Order: | $2592=2^{5} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |